Small-angle impurity scattering and the spin Hall conductivity in two-dimensional semiconductor systems

An arbitrarily small concentration of impurities can affect the spin Hall conductivity in a two-dimensional semiconductor system. We develop a Boltzmann-like equation that can be used for impurity scattering with an arbitrary angular dependence, and for an arbitrary angular dependence of the spin-orbit field b(k) around the Fermi surface. For a model applicable to a two-dimensional hole system in GaAs, if the impurity scattering is not isotropic, we find that the spin Hall conductivity depends on the derivative of b with respect to the energy, on deviations from a parabolic band structure, and on the angular dependence of the scattering. In principle, the resulting spin Hall conductivity can be larger or smaller than the intrinsic value, and can have an opposite sign. In the limit of small-angle scattering, in a model appropriate for small hole concentrations, where the band is parabolic and b proportional to k(3), the spin Hall conductivity has an opposite sign from the intrinsic value, and has a larger magnitude. Our analysis assumes that the spin-orbit splitting b and the transport scattering rate tau(-1) are both small compared to the Fermi energy, but the method is valid for an arbitrary value of b tau.